Hypotenuse

Definition

The hypotenuse is the longest side of a right triangle, the side opposite the right angle. It is a fundamental concept in trigonometry, as it is one of the three sides of a right triangle that are used to define the trigonometric functions.

5 Must Know Facts For Your Next Test

1. The hypotenuse is always the longest side of a right triangle, and it is the only side that can be used to define all three trigonometric functions: sine, cosine, and tangent.
2. The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
3. The trigonometric functions sine, cosine, and tangent are defined using the ratios of the lengths of the sides of a right triangle, with the hypotenuse being the denominator in the cosine and secant functions.
4. The hypotenuse is an important concept in many applications of trigonometry, such as surveying, navigation, and engineering, where it is used to calculate distances, angles, and other geometric properties.
5. The length of the hypotenuse can be calculated using the Pythagorean Theorem, which states that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

Review Questions

• Explain the relationship between the hypotenuse and the other two sides of a right triangle.
  • The hypotenuse is the longest side of a right triangle and is the side opposite the right angle. According to the Pythagorean Theorem, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. This relationship is fundamental to the definition of the trigonometric functions, as the hypotenuse is used as the denominator in the cosine and secant functions.
• Describe how the hypotenuse is used to define the trigonometric functions.
  • The hypotenuse is a critical component in the definition of the three primary trigonometric functions: sine, cosine, and tangent. The sine function is defined as the ratio of the length of the opposite side to the length of the hypotenuse, the cosine function is defined as the ratio of the length of the adjacent side to the length of the hypotenuse, and the tangent function is defined as the ratio of the length of the opposite side to the length of the adjacent side. The hypotenuse is the only side that is used in the definitions of all three trigonometric functions.
• Analyze the importance of the hypotenuse in real-world applications of trigonometry.
  • The hypotenuse is an essential concept in many practical applications of trigonometry, such as surveying, navigation, and engineering. In surveying, the hypotenuse is used to calculate distances between points, which is crucial for mapping and land measurement. In navigation, the hypotenuse is used to determine the distance and direction of travel, which is essential for safe and efficient travel by air, sea, or land. In engineering, the hypotenuse is used to calculate the length of structural elements, such as beams and trusses, which is necessary for ensuring the structural integrity of buildings and other infrastructure. The ability to accurately calculate the length of the hypotenuse is a fundamental skill in these and many other fields that rely on the principles of trigonometry.

Related terms

Adjacent Side: The side of a right triangle that is adjacent to the angle being considered, forming one of the two sides that meet at the right angle.

Opposite Side:

The side of a right triangle that is opposite the angle being considered, forming the other side that meets at the right angle.

Pythagorean Theorem:

A fundamental theorem in geometry that states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.